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Notation: For a group $G$, we write $G^n$ to denote the $n$-fold direct product of $G$ with itself.

Problem set up:

Consider, for a finite group $G$, and $n > 1$, the set $Q(G)_n$ of all isomorphism classes of groups of the form $\frac{G^{n}}{H_{n-1}}$, for any normal subgroup $H_{n-1}$ of $G^n$ isomorphic to $G^{n-1}$.

Note that $Q(G)_1 \subset Q(G)_2 \subset \cdots$, and so for finite groups $G$, the sequence stabilises at some $N$ - i.e. we have $Q(G)_n =Q(G)_{n+1}$ for all $n \geq N$. We call the sequence $Q(G)_1, \cdots, Q(G)_N$ the quotient series of $G$.

We recall that by the classification theorem for finite abelian groups, we can write any finite abelian group as $\bigotimes_{i = 0}^M \mathbb Z_{p_{i}^{k_i}}$ for primes $p_i$ and positive integers $k_i$.

Question: Let $G = \bigotimes_{i = 0}^M \mathbb Z_{p_{i}^{k_i}}$ be a finite abelian group. What is the quotient series of $G$?

Notation: For a group $G$, we write $G^n$ to denote the $n$-fold direct product of $G$ with itself.

Problem set up:

Consider, for a finite group $G$, and $n > 1$, the set $Q(G)_n$ of all isomorphism classes of groups of the form $\frac{G^{n}}{H_{n-1}}$, for any normal subgroup $H_{n-1}$ of $G^n$ isomorphic to $G^{n-1}$.

Note that $Q(G)_1 \subset Q(G)_2 \subset \cdots$, and so for finite groups $G$, the sequence stabilises at some $N$ - i.e. we have $Q(G)_n =Q(G)_{n+1}$ for all $n \geq N$. We call the sequence $Q(G)_1, \cdots, Q(G)_N$ the quotient series of $G$.

We recall that by the classification theorem for finite abelian groups, we can write any finite abelian group as $\bigoplus_{i = 0}^M \mathbb Z_{p_{i}^{k_i}}$ for primes $p_i$ and positive integers $k_i$.

Question: Let $G = \bigoplus_{i = 0}^M \mathbb Z_{p_{i}^{k_i}}$ be a finite abelian group. What is the quotient series of $G$?

Notation: For a group $G$, we write $G^n$ to denote the $n$-fold direct product of $G$ with itself.

Problem set up:

Consider, for a finite group $G$, and $n > 1$, the set $Q(G)_n$ of all isomorphism classes of groups of the form $\frac{G^{n}}{H_{n-1}}$, for any normal subgroup $H_{n-1}$ of $G^n$ isomorphic to $G^{n-1}$.

Note that $Q(G)_1 \subset Q(G)_2 \subset \cdots$, and so for finite groups $G$, the sequence stabilises at some $N$ - i.e. we have $Q(G)_n =Q(G)_{n+1}$ for all $n \geq N$. We call the sequence $Q(G)_1, \cdots, Q(G)_N$ the quotient series of $G$.

Now by the classification theorem for finite abelian groups, we can write any finite abelian group as $\bigotimes_{i = 0}^M \mathbb Z_{p_{i}^{k_i}}$ for primes $p_i$ and positive integers $k_i$. We note that for finite abelian groups $G$, $Q(G)_n$ consists also of finite abelian groups for each $n$.

Question: Let $G = \bigotimes_{i = 0}^M \mathbb Z_{p_{i}^{k_i}}$ be a finite abelian group. What is the quotient series of $G$?

Notation: For a group $G$, we write $G^n$ to denote the $n$-fold direct product of $G$ with itself.

Problem set up:

Consider, for a finite group $G$, and $n > 1$, the set $Q(G)_n$ of all isomorphism classes of groups of the form $\frac{G^{n}}{H_{n-1}}$, for any normal subgroup $H_{n-1}$ of $G^n$ isomorphic to $G^{n-1}$.

Note that $Q(G)_1 \subset Q(G)_2 \subset \cdots$, and so for finite groups $G$, the sequence stabilises at some $N$ - i.e. we have $Q(G)_n =Q(G)_{n+1}$ for all $n \geq N$. We call the sequence $Q(G)_1, \cdots, Q(G)_N$ the quotient series of $G$.

We recall that by the classification theorem for finite abelian groups, we can write any finite abelian group as $\bigotimes_{i = 0}^M \mathbb Z_{p_{i}^{k_i}}$ for primes $p_i$ and positive integers $k_i$.

Question: Let $G = \bigotimes_{i = 0}^M \mathbb Z_{p_{i}^{k_i}}$ be a finite abelian group. What is the quotient series of $G$?